1. trees does the network shown in figure (a) have? (b) How many different spanning. trees does the network shown in figure (b) have?

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1 2/28/18, 8:24 M 1. (a) ow many different spanning trees does the network shown in figure (a) have? (b) ow many different spanning trees does the network shown in figure (b) have? L K M P N O L K M P N O J J (a) (b) (a) The network shown in figure (a) has (b) The network shown in figure (a) has possible spanning tree(s). possible spanning tree(s). 2. truck must deliver furniture to stores located in 5 different cities,,,, and. The truck must start and end its route at. The time (in hours) for travel between the cities is given in the graph to the right. ind the cheapest-link tour for this TSP and give its cost in hours ind the cheapest-link tour. ind the cost for this tour. hours 3. n each case, find the value of N. (a) K N has 6 distinct amilton circuits. (b) K N has 66 edges. (c) K N has 19,900 edges. (a) N = (b) N = (c) N = Page 1 of 14

2 2/28/18, 8:24 M 4. ind an optimal tour for the TSP (traveling salesman problem) to the right, and give its cost ind an optimal tour using the brute-force algorithm. What is the cost of the optimal tour? 5. Use Kruskal's algorithm to find the maximum spanning tree for the weighted graph. ive the total weight of the maximum spanning tree Which of the following trees matches the shape of the maximum spanning tree?.... What is the total weight of the maximum spanning tree? The total weight is. Page 2 of 14

3 2/28/18, 8:24 M Use Kruskal's lgorithm to find the minimum spanning tree for the weighted graph. ive the total weight of the minimum spanning tree Which of the following trees matches the shape of the minimum spanning tree?.... What is the total weight of the minimum spanning tree? The total weight is. 7. Use Kruskal's algorithm to find the maximum spanning tree for the weighted graph. ive the total weight of the maximum spanning tree Which of the following trees matches the shape of the maximum spanning tree?.... What is the total weight of the maximum spanning tree? The total weight is. Page 3 of 14

4 2/28/18, 8:24 M 8. onsider the network shown to the right. (a) ow many degrees of separation are there between and? (b) ow many degrees of separation are there between and? (c) ow many degrees of separation are there between and J? J K N M L (a) There is(are) degree(s) of separation between and. (b) There is(are) degree(s) of separation between and. (c) There is(are) degree(s) of separation between and J. Page 4 of 14

5 2/28/18, 8:24 M 9. Use the accompanying network figures to complete parts (a) through (c) below. 1 lick the icon to view the network figures. (a) ind all the spanning trees of the network shown in figure (a). Select all that apply (b) ind all the spanning trees of the network shown in figure (b). Select all that apply (c) ow many different spanning trees does the network shown in figure (c) have? 1: Networks igure (a) igure (b) J L K igure (c) Page 5 of 14

6 2/28/18, 8:24 M 10. ind the MST of the network shown to the right using Kruskal's algorithm, and give its weight L 8 M 8 N K 8 P O J Which figure represents an MST of the network?.... The weight of the MST using Kruskal's algorithm is. 11. iven below is information about a network. hoose one of the following three options: the network is definitely a tree; the network is definitely not a tree; the network may or may not be a tree (more information is needed). ccompany your answer with a brief explanation for your choice. The network has 19 vertices and 18 edges. hoose the correct answer below.. The network may or may not be a tree because the number of circuits is unknown.. The network is definitely a tree because it satisfies the no-bridges property.. The network is definitely a tree because it satisfies the N + 1 edges property.. The network is definitely not a tree because it violates the single-path property.. The network is definitely not a tree because it violates the N + 1 edges property.. The network is definitely a tree because it satisfies the N 1 edges property.. The network may or may not be a tree because the number of bridges is unknown.. The network is definitely not a tree because it violates the N 1 edges property. 12. Suppose that in solving a TSP you use the nearest-neighbor algorithm and find a nearest-neighbor tour with a total length of 23,220 miles. Suppose that you later find out that the lefngth of an optimal tour is 22,430 miles. What was the relative error of your nearest-neighbor tour? The relative error is %. (Round to the nearest tenth as needed.) Page 6 of 14

7 2/28/18, 8:24 M 13. iven below is information about a network. hoose one of the following three options: the network is definitely a tree; the network is definitely not a tree; the network may or may not be a tree (more information is needed). ccompany your answer with a brief explanation for your choice. The network has 7 vertices ( M through S ), and there is only one path connecting N and R. hoose the correct answer below.. The network is definitely not a tree because it violates the single-path property.. The network is definitely not a tree because it has more edges than vertices.. The network may or may not be a tree because the other vertices may be connected by single or multiple paths.. The network is definitely a tree because it satisfies the single-path property.. The network is definitely a tree because it satisfies the all-bridges property.. The network may or may not be a tree because the path connecting N and R is a bridge, but the other edges may or may not be bridges. 14. robotic laser must drill holes on five sites (,,,, and ) in a microprocessor chip. t the end, the laser must return to its starting position and start all over. The accompanying table shows the time (in seconds) it takes the laser arm to move from one site to another. n this TSP, a tour is a sequence of drilling locations starting and ending at. omplete parts (a) and (b) below. (a) ind the cheapest-link tour and its length. Starting at, the cheapest-link tour is. * * * * * The length of this tour is (Type an integer or a decimal.) seconds. (b) iven that the tour,,,,, is an optimal tour, find the relative error of the cheapest-link tour found in (a). ε = % (Round to two decimal places as needed.) Page 7 of 14

8 2/28/18, 8:24 M 15. or the graph on the right, complete parts (a) through (c) below. (a) ind three different amilton circuits. Select all that apply..,,,,,,,.,,,,,,,.,,,,,,,.,,,,,,,.,,,,,,,.,,,,,,, (b) ind a amilton path that starts at and ends at. (c) ind a amilton path that starts at and ends at. 16. Suppose,,,,,,,, is a amilton circuit in a graph. omplete parts (a) through (c) below. (a) ind the number of vertices in the graph. (b) Write the amilton circuit using as the starting/ending vertex. (c) ind two different amilton paths in the graph that start at. Select all that apply..,,,,,,,.,,,,,,,.,,,,,,,.,,,,,,, Page 8 of 14

9 2/28/18, 8:24 M 17. or the weighted graph to the right, complete parts (a) through (c) (a) ind a amilton path that starts at and ends at, and give its weight. dentify a amilton path that starts at and ends at. hoose the correct answer below.,,,,,.,,,,,.,,,,,.,,,,, The weight of this path is. (b) ind a second amilton path that starts at and ends at, and give its weight. dentify a second amilton path that starts at and ends at. The weight of this path is. (c) ind the optimal (least weight) amilton path that starts at and ends at, and give its weight. dentify the optimal amilton path that starts at and ends at. hoose the correct answer below.,,,,,.,,,,,.,,,,,.,,,,,.,,,,,.,,,,,.,,,,,.,,,,, The weight of this path is. Page 9 of 14

10 2/28/18, 8:24 M 18. truck must deliver furniture to stores located in 5 cities,,,, and. The truck must start and end its route at. The time (in hours) for travel between the cities is given in the graph to the right. omplete parts (a) and (b) below a) ind the nearest-neighbor tour starting at. hoose the correct answer below. b) ind the nearest-neighbor tour starting at, and give the answer using as the starting/ending city. hoose the correct answer below. 19. (a) ow many edges are there in K 6? (b) ow many edges are there in K 8? (c) f the number of edges in K 32 is x, and the number of edges in K 33 is y, what is the value of y x? (a) The number of edges in K 6 is. (b) The number of edges in K 8 is. (c) The value of y x is Use Kruskal's lgorithm to find the minimum spanning tree for the weighted graph. ive the total weight of the minimum spanning tree Which of the following trees matches the shape of the minimum spanning tree?.... What is the total weight of the minimum spanning tree? The total weight is. Page 10 of 14

11 2/28/18, 8:24 M 21. truck must deliver furniture to stores located in 5 cities,,,, and. The truck must start and end its route at. The time (in hours) for travel between the cities is given in the graph to the right. ind the repetitive nearest-neighbor tour, and give its cost The repetitive nearest-neighbor tour starting with the vertex is. The cost of this tour is hours. 22. ow many different spanning trees do the networks shown have? (a) (b) J J M L K M L K (a) The network has (b) The network has different spanning trees. different spanning trees. Page 11 of 14

12 2/28/18, 8:24 M ,,,,, ,,,, Page 12 of 14

13 2/28/18, 8:24 M 9..,.,..,.,., The network is definitely a tree because it satisfies the N 1 edges property The network may or may not be a tree because the other vertices may be connected by single or multiple paths. 14.,,,,, ,,,,,,,,.,,,,,,,,.,,,,,,,,,,,,,,,,,,, 16. 8,,,,,,,,.,,,,,,,,.,,,,,,, Page 13 of 14

14 2/28/18, 8:24 M 17..,,,,, 30,,,,, 28.,,,,, ,,,,,,,,,, ,,,,, Page 14 of 14

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